Description: Version of equsalh with a disjoint variable condition, which does not require ax-13 . Remark: this is the same as equsalhw . TODO: delete after moving the following paragraph somewhere.
Remarks: equsexvw has been moved to Main; Theorem ax13lem2 has a DV version which is a simple consequence of ax5e ; Theorems nfeqf2 , dveeq2 , nfeqf1 , dveeq1 , nfeqf , axc9 , ax13 , have dv versions which are simple consequences of ax-5 . (Contributed by BJ, 14-Jun-2019) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | bj-equsalhv.nf | ⊢ ( 𝜓 → ∀ 𝑥 𝜓 ) | |
| bj-equsalhv.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | ||
| Assertion | bj-equsalhv | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-equsalhv.nf | ⊢ ( 𝜓 → ∀ 𝑥 𝜓 ) | |
| 2 | bj-equsalhv.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| 3 | 1 | nf5i | ⊢ Ⅎ 𝑥 𝜓 |
| 4 | 3 2 | equsalv | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 ) |